The trace is just the mathematical description of how you get from quantum to classical. The TL;DR is that a proper subset of a quantum system behaves, when considered separately from the rest of the system, as if it were a classical system even though it really isn't. To see the details, read this:
Just to be clear where I'm coming from, I have taken a course on quantum information theory, so I'm not completely lost here. But some things I for-got and somethings I never-got.
And in particular, the (separate) course on qft lost me at hello, so I never got the chance to incorporate that into my worldview.
In that case: if you look at an entire macroscopic system, isolated from everything else, before and after a measurement, you will find that the total entropy of the system has not changed. It can't because the evolution of an isolated system is always unitary.
If you trace over any degree of freedom of the system (i.e. separate the system into two parts, which are usually the "particle being measured" and everything else) then what you will find is that after the measurement the degree of freedom over which you traced has a negative entropy, and the rest of the system now has a positive entropy (which is the source of the apparent randomness), and that all of the degrees of freedom of the rest of the system (in the preferred basis as determined by decoherence) are now in classical correlation with each other. That is how you make "copies" of (classical) information.
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.79...
For some additional background:
https://adami.natsci.msu.edu/blog/2014/6/26/on-quantum-measu...